Differentiation rules explained

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

Elementary rules of differentiation

Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined^{[1] [2]} — including the case of complex numbers (C).^[3]

Constant term rule

For any value of

c , where c \in R , if f(x) is the constant function given by f(x) = c , then (df)/(dx) = 0 .^[4]

Proof

Let

c \in R and f(x) = c . By the definition of the derivative,\begin{align} f'(x) & = \lim _{h} (f(x + h) - f(x))/(h) \\ & = \lim _{h} ((c) - (c))/(h) \\ & = \lim _{h} (0)/(h) \\ & = \lim _{h} 0 \\ & = 0 \end{align}

This shows that the derivative of any constant function is 0.

Intuitive (geometric) explanation

The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero.

In other words, the value of the constant function, y, will not change as the value of x increases or decreases.

Differentiation is linear

See main article: Linearity of differentiation.

For any functions

f and g and any real numbers a and b , the derivative of the function h(x) = af(x) + bg(x) with respect to x is: h'(x) = a f'(x) + b g'(x).

In Leibniz's notation this is written as:$$\backslash frac\; =\; a\backslash frac\; +b\backslash frac.$$

Special cases include:

• The constant factor rule $$(af)\text{'}\; =\; af\text{'}$$
• The sum rule $$(f\; +\; g)\text{'}\; =\; f\text{'}\; +\; g\text{'}$$
• The difference rule $$(f\; -\; g)\text{'}\; =\; f\text{'}\; -\; g\text{'}.$$

The product rule

See main article: Product rule.

For the functions

f and g , the derivative of the function h(x) = f(x) g(x) with respect to x is$$h\text{'}(x)\; =\; (fg)\text{'}(x)\; =\; f\text{'}(x)\; g(x)\; +\; f(x)\; g\text{'}(x).$$In Leibniz's notation this is written$$\backslash frac\; =\; g\; \backslash frac\; +\; f\; \backslash frac.$$

The chain rule

See main article: Chain rule.

The derivative of the function

h(x) = f(g(x)) is$$h\text{'}(x)\; =\; f\text{'}(g(x))\backslash cdot\; g\text{'}(x).$$

In Leibniz's notation, this is written as:$$\backslash frach(x)\; =\; \backslash left.\backslash fracf(z)\backslash right|\_\backslash cdot\; \backslash fracg(x),$$often abridged to$$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac.$$

Focusing on the notion of maps, and the differential being a map

D , this is written in a more concise way as:$$[\backslash text\{D\}\; (f\backslash circ\; g)]\_x\; =\; [\backslash text\{D\}\; f]\_\; \backslash cdot\; [\backslash text\{D\}g]\_x\backslash ,.$$

The inverse function rule

See main article: Inverse function rule.

If the function has an inverse function, meaning that

g(f(x)) = x and f(g(y)) = y, then$$g\text{'}\; =\; \backslash frac.$$

In Leibniz notation, this is written as$$\backslash frac\; =\; \backslash frac.$$

Power laws, polynomials, quotients, and reciprocals

The polynomial or elementary power rule

See main article: Power rule.

If

f(x) = x ^{r} , for any real number r ≠ 0 , then f'(x) = rx ^{r-1} .

When

r = 1 , this becomes the special case that if f(x) = x, then f'(x) = 1 .

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

The reciprocal rule

See main article: Reciprocal rule. The derivative of

h(x) = (1)/(f(x)) for any (nonvanishing) function is:h'(x) =- (f'(x))/((f(x)) ^{2}) wherever is non-zero.

In Leibniz's notation, this is written

(d( 1 /f))/(dx) =- (1)/(f ^{2}) (df)/(dx) .

The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.

The quotient rule

See main article: Quotient rule. If and are functions, then:

\left( (f)/(g) \right)' = (f'g - g'f)/(g ^{2}) wherever is nonzero.

This can be derived from the product rule and the reciprocal rule.

Generalized power rule

See main article: Power rule.

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions and ,

(f ^{g)'} = \left(e ^{gln} \right)' = f ^{g\left(f'{g} \over f} + g' ln f\right), wherever both sides are well defined.

Special cases

• If $f(x)=x^a\backslash !$, then $f\text{'}(x)=ax^$ when is any non-zero real number and is positive.
• The reciprocal rule may be derived as the special case where $g(x)=-1\backslash !$.

Derivatives of exponential and logarithmic functions

(d)/(dx) \left(c ^{ax} \right) = {ac ^{ax} ln c } , c > 0 the equation above is true for all, but the derivative for yields a complex number.(d)/(dx) \left(e ^{ax} \right) = ae ^{ax} (d)/(dx) \left( log _{c} x\right) = { 1 \over x ln c} , c > 1

the equation above is also true for all , but yields a complex number if .

(d)/(dx) \left( ln x\right) = { 1 \over x} , x > 0 . (d)/(dx) \left( ln |x|\right) = { 1 \over x} , x ≠ 0 . (d)/(dx) \left( W(x)\right) = { 1 \over {x + e ^{W(x)} }},\qquad x > -.\qquadwhere W(x) is the Lambert W function(d)/(dx) \left( x ^{x} \right) = x ^{x(1+ln} x). (d)/(dx) \left( f(x) \right ) = g(x)f(x) ^{g(x)-1} (df)/(dx) + f(x) ^{g(x)} ln {( f(x) )} (dg)/(dx) , if f(x) > 0 , andif (df)/(dx) and (dg)/(dx) exist. (d)/(dx) \left( f _{1} (x) ^{f _{2} (x) ^{\left ( ... \right ) ^{f _{n} (x)}}} \right ) = \left [\sum\limits ^{n} _{k =1} (\partial)/(\partial x _{k}) \left( f _{1} (x ^{f _{2} (x ^{\left ( ... \right ) ^{f _{n} (x _{n)}}} _{2 )}} _{1 )} \right ) \right ] r \vert _{x _{1} = x _{2} = ... = x _{n} = x} , if f _{i<n} (x) > 0 and (df _{i})/(dx) exists.

Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

( ln f)' = (f')/(f) wherever is positive.

Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.

Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.

Derivatives of trigonometric functions

See main article: Differentiation of trigonometric functions.

width=50%	(d)/(dx) \sin x = \cos x	width=50%	(d)/(dx) \arcsin x = (1)/(\sqrt{ 1- x ^{2}) }
(d)/(dx) \cos x = - \sin x	(d)/(dx) \arccos x =- (1)/(\sqrt{ 1- x ^{2}) }
(d)/(dx) \tan x = \sec ^{2} x = (1)/(\cos ^{2} x) = 1 + \tan ^{2} x	(d)/(dx) \arctan x = (1)/(1+ x ^{2})
(d)/(dx) \csc x = - \csc{x}\cot{x}	(d)/(dx) \operatorname{arccsc} x =- (1)/(|x|\sqrt{x ^{2} -1) }
(d)/(dx) \sec x = \sec{x}\tan{x}	(d)/(dx) \operatorname{arcsec} x = (1)/(|x|\sqrt{x ^{2} -1) }
(d)/(dx) \cot x = - \csc ^{2} x =- (1)/(\sin ^{2} x) = -1 - \cot ^{2} x	(d)/(dx) \operatorname{arccot} x = - { 1 \over 1 + x ^{2}}

The derivatives in the table above are for when the range of the inverse secant is [ 0 ,\pi] and when the range of the inverse cosecant is \left[ - (\pi)/(, 2) (\pi)/(2) \right].

It is common to additionally define an inverse tangent function with two arguments,

\arctan(y,x). Its value lies in the range [ - \pi,\pi] and reflects the quadrant of the point (x,y). For the first and fourth quadrant (i.e. x > 0 ) one has \arctan(y, x> 0 ) = \arctan(y/x). Its partial derivatives are(\partial \arctan(y,x))/(\partial y) = (x)/(x ^{2} + y ^{2}) and (\partial \arctan(y,x))/(\partial x) = (- y)/(x ^{2} + y ^{2}) .

Derivatives of hyperbolic functions

width=50%	(d)/(dx) \sinh x = \cosh x	(d)/(dx) \operatorname{arsinh}x = (1)/(\sqrt{ 1+ x ^{2}) }
(d)/(dx) \cosh x = \sinh x	(d)/(dx) \operatorname{arcosh} x = { (1)/(\sqrt{x ^{2-1}) }}
(d)/(dx) \tanh x = {\operatorname{sech} ^{2} x} = 1 - \tanh ^{2} x	(d)/(dx) \operatorname{artanh} x = (1)/(1- x ^{2})
(d)/(dx) \operatorname{csch} x = - \operatorname{csch}{x}\coth{x}	(d)/(dx) \operatorname{arcsch} x =- (1)/(|x|\sqrt{ 1+ x ^{2}) }
(d)/(dx) \operatorname{sech} x = - \operatorname{sech}{x}\tanh{x}	(d)/(dx) \operatorname{arsech} x =- (1)/(x\sqrt{ 1- x ^{2}) }
(d)/(dx) \coth x = - \operatorname{csch} ^{2} x = 1 - \coth ^{2} x	(d)/(dx) \operatorname{arcoth} x = (1)/(1- x ^{2})

See Hyperbolic functions for restrictions on these derivatives.

Derivatives of special functions

Gamma function

\Gamma(x) = \int ^{inf ty} _{0} t ^{x-1} e ^{-t} dt \begin{align} \Gamma'(x) & = \int ^{inf ty} _{0} t ^{x-1} e ^{-t} ln tdt \\ & = \Gamma(x) \left(\sum ^{inf ty} _{n =1} \left( ln \left( 1 + \dfrac{ 1 }{n}\right) - \dfrac{ 1 }{x + n}\right) - \dfrac{ 1 }{x}\right) \\ & = \Gamma(x) \psi(x) \end{align} with \psi(x) being the digamma function, expressed by the parenthesized expression to the right of \Gamma(x) in the line above.

Riemann zeta function

\zeta(x) = \sum ^{inf ty} _{n =1} (1)/(n ^{x}) \begin{align} \zeta'(x) & = - \sum ^{inf ty} _{n =1} (ln n)/(=- n ^{x}) (ln2)/(2 ^{x}) - (ln3)/(3 ^{x}) - (ln4)/(4 ^{x}) - … \\ & = - \sum _{p} (p ^{-x} ln p)/(( 1- p ^{-x} ) ^{2}) \prod _{q} (1)/(1- q ^{-x}) \end{align}

Derivatives of integrals

Suppose that it is required to differentiate with respect to x the function

F(x) = \int ^{b(x)} _{a(x)} f(x,t)dt,

where the functions

f(x,t) and (\partial)/(\partial x) f(x,t) are both continuous in both t and x in some region of the (t,x) plane, including a(x)\leq t\leq b(x), x _{0\leq} x\leq x _{1} , and the functions a(x) and b(x) are both continuous and both have continuous derivatives for x _{0\leq} x\leq x _{1} . Then for x _{0\leq} x\leq x _{1} :F'(x) = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int ^{b(x)} _{a(x)} (\partial)/(\partial x) f(x,t) dt.

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

Derivatives to nth order

Some rules exist for computing the -th derivative of functions, where is a positive integer. These include:

Faà di Bruno's formula

See main article: Faà di Bruno's formula. If and are -times differentiable, then$$\backslash frac\; [f(g(x))]=\; n!\; \backslash sum\_\; f^(g(x))\; \backslash prod\_^n\; \backslash frac\; \backslash left(g^(x)\; \backslash right)^$$where $r\; =\; \backslash sum\_^\; k\_m$ and the set

\{k _{m\}} consists of all non-negative integer solutions of the Diophantine equation $\backslash sum\_^\; m\; k\_m\; =\; n$.

General Leibniz rule

See main article: General Leibniz rule. If and are -times differentiable, then$$\backslash frac[f(x)g(x)]\; =\; \backslash sum\_^\; \backslash binom\; \backslash frac\; f(x)\; \backslash frac\; g(x)$$

Sources and further reading

These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in:

• Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, .
• The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, .
• Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,
• NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, .

External links

• Derivative calculator with formula simplification

Notes and References

1. Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, .
2. Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, .
3. Complex Variables, M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009,
4. Web site: Differentiation Rules . University of Waterloo – CEMC Open Courseware . 3 May 2022.